Page "Prime number theorem" Paragraph 12

Without doubt, the single most significant paper concerning the distribution of prime numbers was Riemann's 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject.

Riemann introduced revolutionary ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable.

In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π ( x ) originates.

Extending these deep ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin and appeared in the same year ( 1896 ).

Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ ( s ) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0.